RATIONAL CURVES IN A QUADRIC THREEFOLD VIA AN SL(2, C)-REPRESENTATION

초록

In this paper, we regard the smooth quadric threefold $Q_3$ as Lagrangian Grassmannian and search for fixed rational curves of low degree in $Q_3$ with respect to a torus action, which is the maximal subgroup of the special linear group $\text{SL}(2,\mathbb{C})$. Most of them are confirmations of very well-known facts. If the degree of a rational curve is $3$, it is confirmed using the Lagrangian's geometric properties that the moduli space of twisted cubic curves in $Q_3$ has a specific projective bundle structure. From this, we can immediately obtain the cohomology ring of the moduli space.

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